5.1 Continuous Probability Functions
Probability density function (pdf) f(x):
- f(x) ≥ 0
- The total area under the curve f(x) is one.
Cumulative distribution function (cdf): P(X ≤ x)
5.2 The Uniform Distribution
X = a real number between a and b (in some instances, X can take on the values a and b). a = smallest X; b = largest X
X ~ U(a, b)
The mean is μ=a+b2.
The standard deviation is σ=√(b – a)212.
Probability density function: f(x)=1b−a for a≤X≤b
Area to the left of x: P(X < x) = (x – a)(1b−a)
Area to the right of x: P(X > x) = (b – x)(1b−a)
Area between c and d: P(c < x < d) = (base)(height) = (d – c)(1b−a)
Uniform: X ~ U(a, b) where a < x < b
- pdf: f(x)=1b−a for a ≤ x ≤ b
- cdf: P(X ≤ x) = x−ab−a
- mean µ = a+b2
- standard deviation σ =√(b−a)212
- P(c < X < d) = (d – c)(1b–
5.3 The Exponential Distribution (Optional)
Exponential: X ~ Exp(m) where m = the decay parameter
- pdf: f(x) = me(–mx) where x ≥ 0 and m > 0
- cdf: P(X ≤ x) = 1 – e(–mx)
- mean µ =
- standard deviation σ = µ
- percentile k: k =
- Additionally:
- P(X > x) = e(–mx)
- P(a < X < b) = e(–ma) – e(–mb)
- Memoryless property: P(X > x + k|X > x) = P (X > k)
- Poisson probability: with mean λ
- k! = k*(k−1)*(k−2)*(k−3)*…3*2*1